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Geometric Invariant Theory : Ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 2. Folge - David Mumford

Geometric Invariant Theory

Ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 2. Folge

Hardcover

Published: March 1994
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"Geometric Invariant Theory" by Mumford/Fogarty (the firstedition was published in 1965, a second, enlarged editonappeared in 1982) is the standard reference on applicationsof invariant theory to the construction of moduli spaces.This third, revised edition has been long awaited for by themathematical community. It is now appearing in a completelyupdated and enlarged version with an additional chapter onthe moment map by Prof. Frances Kirwan (Oxford) and a fullyupdated bibliography of work in this area.The book deals firstly with actions of algebraic groups onalgebraic varieties, separating orbits by invariants andconstructionquotient spaces; and secondly with applicationsof this theory to the construction of moduli spaces.It is a systematic exposition of the geometric aspects ofthe classical theory of polynomial invariants.

Preliminariesp. 1
Definitionsp. 2
First propertiesp. 4
Good and bad actionsp. 9
Further propertiesp. 13
Resume of some results of Grothendieckp. 19
Fundamental theorems for the actions of reductive groupsp. 24
Definitionsp. 24
The affine casep. 27
Linearization of an invertible sheafp. 30
The general casep. 36
Functional propertiesp. 44
Analysis of stabilityp. 48
A numeral criterionp. 48
The flag complexp. 55
Applicationsp. 63
An elementary examplep. 67
Pre-stabilityp. 67
Stabilityp. 72
Further examplesp. 76
Binary quanticsp. 76
Hypersurfacesp. 79
Counter-examplesp. 83
Sequences of linear subspacesp. 86
The projective adjoint actionp. 88
Space curvesp. 89
The problem of moduli - 1st constructionp. 96
General discussionp. 96
Moduli as an orbit spacep. 98
First chern classesp. 104
Utilization of 4.6p. 109
Abelian schemesp. 115
Dualsp. 115
Polarizationsp. 120
Deformationsp. 124
The method of covariants - 2nd constructionp. 127
The techniquep. 127
Moduli as an orbit spacep. 129
The covariantp. 138
Application to curvesp. 142
The moment mapp. 144
Symplectic geometryp. 144
Symplectic quotients and geometric invariant theoryp. 148
Kahler and hyperkahler quotientsp. 152
Singular quotientsp. 156
Geometry of the moment mapp. 160
The cohomology of quotients: the symplectic casep. 164
The cohomology of quotients: the algebraic casep. 172
Vector bundles and the Yang-Mills functionalp. 181
Yang-Mills theory over Riemann surfacesp. 185
Appendix to Chapter 1p. 191
Appendix to Chapter 2p. 202
Appendix to Chapter 3p. 205
Appendix to Chapter 4p. 206
Appendix to Chapter 5p. 217
Appendix to Chapter 7p. 234
Referencesp. 253
Index of definitions and notationsp. 291
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540569633
ISBN-10: 3540569634
Series: Ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 2. Folge
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 294
Published: March 1994
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 24.4 x 16.3  x 2.4
Weight (kg): 0.62
Edition Number: 3
Edition Type: Revised